Degradation properties and metabolic activity of alginate and chitosan polyelectrolytes for drug delivery and tissue engineering applications

Vincenzo Guarino; Tania Caputo; Rosaria Altobelli; Luigi Ambrosio; Vincenzo Guarino; Tania Caputo; Rosaria Altobelli; Luigi Ambrosio

doi:10.3934/matersci.2015.4.497

AIMS Materials Science

2015, Volume 2, Issue 4: 497-502. doi: 10.3934/matersci.2015.4.497

Previous Article Next Article

Review Topical Sections

Degradation properties and metabolic activity of alginate and chitosan polyelectrolytes for drug delivery and tissue engineering applications

Institute of Polymers, Composites and Biomaterials, National Research Council of Italy, Mostra d'Oltremare, Pad.20, V.le J.F. Kennedy 54, Naples, Italy

Received: 12 October 2015 Accepted: 12 November 2015 Published: 23 November 2015

Polysaccharides are long monosaccharide units which are emerging as promising materials for tissue engineering and drug delivery applications due to their biocompatibility, mostly good availability and tailorable properties, by to the wide possibility to modify chemical composition, structure—i.e., linear chain or branching—and polymer source (animals, plants, microorganisms). For their peculiar behaviour as polyelectrolites, polysaccharides have been applied in various forms, such as injectable hydrogels or porous and fibrous scaffolds—alone or in combination with other natural or synthetic polymers—to design bioinspired platforms for the regeneration of different tissues (i.e., blood vessels, myocardium, heart valves, bone, articular and tracheal cartilage, intervertebral discs, menisci, skin, liver, skeletal muscle, neural tissue, urinary bladder) as well as for encapsulation and controlled delivery of drugs for pharmaceutical devices. In this paper, we focus on the pH sensitive response and degradation behaviour of negative (i.e., alginate) and positive (i.e., chitosan) charged polysaccharides in order to discuss the differences in terms of metabolic activity of polyelectrolytes with different ionic strength for their use in drug delivery and tissue engineering area.

Keywords:

Citation: Vincenzo Guarino, Tania Caputo, Rosaria Altobelli, Luigi Ambrosio. Degradation properties and metabolic activity of alginate and chitosan polyelectrolytes for drug delivery and tissue engineering applications[J]. AIMS Materials Science, 2015, 2(4): 497-502. doi: 10.3934/matersci.2015.4.497

Related Papers:

[1]	Abdul M. Siddiqui, Getinet A. Gawo, Khadija Maqbool . On slip of a viscous fluid through proximal renal tubule with linear reabsorption. AIMS Biophysics, 2021, 8(1): 80-102. doi: 10.3934/biophy.2021006
[2]	Boukedjane Mouloud, Bahi Lakhdar . Characterizing pulsatile blood flow in a specific carotid bifurcation: insights into hemodynamics and rheology models. AIMS Biophysics, 2023, 10(3): 281-316. doi: 10.3934/biophy.2023019
[3]	Daniel N. Riahi, Saulo Orizaga . On modeling arterial blood flow with or without solute transport and in presence of atherosclerosis. AIMS Biophysics, 2024, 11(1): 66-84. doi: 10.3934/biophy.2024005
[4]	Mohamed A. Elblbesy, Bothaina A. Kandil . Hemorheological measurements over the shear-thinning regime. In vitro comparative study for hyperglycemia. AIMS Biophysics, 2024, 11(2): 121-129. doi: 10.3934/biophy.2024008
[5]	David Gosselin, Maxime Huet, Myriam Cubizolles, David Rabaud, Naceur Belgacem, Didier Chaussy, Jean Berthier . Viscoelastic capillary flow: the case of whole blood. AIMS Biophysics, 2016, 3(3): 340-357. doi: 10.3934/biophy.2016.3.340
[6]	Bradley Vis, Jonathan J. Powell, Rachel E. Hewitt . Imaging flow cytometry methods for quantitative analysis of label-free crystalline silica particle interactions with immune cells. AIMS Biophysics, 2020, 7(3): 144-166. doi: 10.3934/biophy.2020012
[7]	Giulia Rigamonti, Marco Guardamagna, Sabina Merlo . Non-contact reflectometric readout of disposable microfluidic devices by near infra-red low-coherence interferometry. AIMS Biophysics, 2016, 3(4): 585-595. doi: 10.3934/biophy.2016.4.585
[8]	Wenyuan Xie, Jason Wei Jun Low, Arunmozhiarasi Armugam, Kandiah Jeyaseelan, Yen Wah Tong . Regulation of Aquaporin Z osmotic permeability in ABA tri-block copolymer. AIMS Biophysics, 2015, 2(3): 381-397. doi: 10.3934/biophy.2015.3.381
[9]	Mohamed A. Elblbesy . Impact of erythrocytes properties on blood surface tension. AIMS Biophysics, 2019, 6(2): 39-46. doi: 10.3934/biophy.2019.2.39
[10]	Masoud Jabbary, Hossein Rajabi . High-resolution phantom of a nephron for radiation nephrotoxicity evaluation in biophysical simulations. AIMS Biophysics, 2022, 9(2): 147-160. doi: 10.3934/biophy.2022013

Abstract

1. Introduction

The microstructure of portable artificial kidneys is responsible for the blood filtration and urine formation (mechanical process of biofluid) which acts as a micromachine due to very small size but do the complex process of reabsorption, filtration, excretion and secretion in its different parts. Kidneys contain renal speck and renal tubule; renal speck is further split into glomerulus and Bowman's capsule where blood is filtered through the glomerulus circulates and filtered blood then comes to the renal tubule where reabsorption takes place. Within the nephron, glomerular purification comprises on reflexive movement of blood plasma from glomerulus capillaries to the Bowman's capsule that is easily permeable to H₂O, small solutes like urea, sodium ion, and glucose but not porous to hemoglobin and platelets. The next part of the renal tubule is the proximal convoluted tubule (PCT) where effective and reactive transport take place to reabsorb glucose, sodium chloride, and water from the glomerular filtrate. The current model describes biomechanics of plasma having variable viscosity depending upon the distance from centre to the proximal tube wall that is the main segment of kidneys where constant reabsorption and filtration take place.

Different studies of viscous flow through renal tubule having permeable wall property with constant, linear and nonlinear rate of reabsorption have been discussed. Burgen et. al. [1] introduced a study of viscous flow through kidney with glucose reabsorption and developed mathematical equations for solute reabsorption and used the results to check the high affinity of glucose. The dynamics of biofluid through renal tubule and flow behavior in the presence of constant reabsorption on the wall has been discussed by Macey et. al. [2]. He analysed the viscous flow through infinite cylinder using the Navier Stokes equations with constant and linear reabsorption on the walls of cylinder and noticed the Poiseuille flow for constant reabsorption and for the linear reabsorption the flow lines become flattens instead of curve lines. The dynamical properties of the viscous flow through a permeable tube had been examined by Marshall et. al. [3] and found the exact solution with the help of Navier-Stokes equation for velocity and pressure using the Darcy's law also, the approximate solution for the leakage flux has calculated for the proximal renal tubule. Berkstresser et. al. [4], Tewarson et. al. [5] and Gupta et. al. [6] introduced the numerical and analytic techniques for the solution of mathematical model of viscous flow through renal tubule. Ahmad et al. [7] discussed the exact solution of viscous flow through the renal tube and showed the periodic flow in radial direction of the tube. Muthu et al. [8] presented the finite difference method to find the viscous flow through the cross section of tube with permeable boundary, he found the axial velocity and mean pressure generated by the reabsorption on the wall and flux on the axial direction. Koplik et al. [9], Pozrikidis et al. [10], Kropinski et al. [11] and Haroon et al [12] obtained the results for volume flow rate, glucose reabsorption, velocity and shear stress for Newtonian flow through permeable tube and slit by different methods. They found the slip and no slip effect on the velocity, leakage flux, flow rate and used the resulting formulas to find the reabsorption rate and mean pressure against the fractional reabsorption.

The viscosity of plasma filtered in kidney is not a fixed quantity, but it changes according to the diameter of proximal tubule, also flow resistance in renal tube decreases strongly with reduced diameter of the tube which is less than 0.3mm as mentioned in the study of Gilmer et. al. [13]. The flow resistance of the fluid to be filtered in proximal tube is considerably the case of Poiseuille flow because of pressure gradient which describes that resistance in the flow is low at the centre of tube and high near the tube wall. Few studies have been presented the effect of viscosity on the blood flow through kidney (Lamport et. al. [14],[15], Gaylor et. al. [16] and Omori et. al. [17], Halfenstein et. al. [18], Mukhopadhyay et. al. [19],[20]) but in all these studies only theoretical and experimental data has been used. Recently, Mehboob et. al. [21] presented the effect of variable viscosity and constant reabsorption on the slow viscous flow through narrow straight tubule and found the formulas for the leakage flux, mean pressure drop and flow rate but this problem was formulated in cylindrical coordinate system. There is not a single study available in literature which describes the mathematical formulas of creeping flow having varying viscosity across the height of channel with uniform absorption on the permeable walls of the channel in rectangular coordinate system.

Keeping above studies in mind, it has been assumed that the viscosity is an exponential function of distance from the centre to the boundary and reabsorption on the wall is constant. In this paper to study the effect of exponential type of viscosity for the viscous flow through a kidney, authors have considered the uniform reabsorption on the wall and discussed the problem formulation in section 2, methodology of complex system is included in section 3, mathematical results are presented in fourth section, application of the problem is in section 5, discussion is in section 6 and conclusion is included in section 7.

2. Materials and methods

The present mathematical model suggests that a viscous fluid across a rectangular cross section is flowing under the effect of constant flux applied at the entrance of slit. The -axis locating at the middle of rectangular slit and -axis in the vertical direction of centre line as mentioned in Figure 1, due to symmetric behaviour, the upper half of a rectangular slit has considered to reduce the calculations.

Since the kidney contains the proximal renal tube and movement in this tube is of Poiseuille type Layton et al [22] which requires that axial velocity is highest at the middle of slit that is stated in Eq (7). Since boundaries of renal tubule are permeable and reabsorption takes place through these boundaries at a constant rate which is stated in Eq (8). These hypotheses support to assume that thickness is a decreasing function of distance from base to the boundary, therefore in the current research authors have assumed the Navier-Stokes equation with non-constant viscosity. The current problem of two-dimension, incompressible steady state creeping flow through slit become complicated due to non-constant viscosity.

Figure 1. Schematic diagram of the problem.

DownLoad: Full-Size Img PowerPoint

The suggested model recommends that movement of fluid through a slit satisfy the following flow field.

$V = [u (x, y), v (x, y), 0],$ (1)

where u and v represents horizontal and transverse velocity components.

The fluid flow through the nephron is incompressible i.e density is constant throughout the flow field and satisfy the following continuity equation

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.$ (2)

The creeping viscous flow across a slit suggests that in the absence of inertia term NSE's with variable viscosity satisfy the following set of equations.

$\frac{\partial p}{\partial x} = \frac{\partial}{\partial x} (2 µ (y) \frac{\partial u}{\partial x}) + \frac{\partial}{\partial y} {µ (y) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})},$ (3)

$\frac{\partial p}{\partial y} = \frac{\partial}{\partial x} {µ (y) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})} + \frac{\partial}{\partial y} (2 µ (y) \frac{\partial v}{\partial y}) .$ (4)

2.1. Exponential viscosity

Case I: $µ (y) = e^{- β y}$

Since the viscosity is a decreasing function of radius of tube which is mentioned in Ref. (Gilmer et al [13]& Tripathi et al [23]) therefore, the variable viscosity in the following form has chosen

$µ (y) = e^{- β y}, β > 0.$ (5)

The assumptions of plasma flow through renal proximal tube and constant reabsorption on the permeable boundary satisfy the following boundary conditions

$\frac{\partial u}{\partial y} = 0, v = 0, a t y = 0,$ (6)

$u = 0, v = V_{0}, a t y = H,$ (7)

$Q_{0} = 2 W \int_{0}^{H} u (0, y) d y .$ (8)

Stream function and velocity profile are associated by the following expressions

$u = \frac{\partial ψ}{\partial y}, v = - \frac{\partial ψ}{\partial x} .$ (9)

The above expressions of stream function and Eqs (3,4) give the following form

$\frac{\partial p}{\partial x} = 2 e^{- β y} \frac{\partial^{3} ψ}{\partial x^{2} \partial y} + \frac{\partial}{\partial y} {e^{- β y} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}})},$ (10)

$\frac{\partial p}{\partial y} = e^{- β y} \frac{\partial}{\partial x} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}}) + 2 \frac{\partial}{\partial y} {e^{- β y} (- \frac{\partial^{2} ψ}{\partial x \partial y})} .$ (11)

Eliminating pressure gradient from Eqs (10,11), one can write the following expression.

$\nabla^{4} ψ - β^{2} E^{2} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (12)

where $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ a and $E^{2} = \frac{\partial^{2}}{\partial x^{2}} - \frac{\partial^{2}}{\partial y^{2}}$ .

Boundary conditions in terms of stream function are given as follows

$\frac{\partial^{2} ψ}{\partial y^{2}} = 0, \frac{\partial ψ}{\partial x} = 0, a t y = 0,$ (13)

$\frac{\partial ψ}{\partial y} = 0, \frac{\partial ψ}{\partial x} = - V_{0}, a t y = H,$ (14)

$ψ (0, 0) = 0, a n d ψ (0, H) = \frac{Q_{0}}{2 W} .$ (15)

The non-dimensional quantities are chosen from Ref. [12]

$\begin{array}{l} x^{*} = \frac{x}{L}, y^{*} = \frac{y}{L}, V_{0}^{*} = \frac{V_{0}}{U_{0}}, ψ^{*} = \frac{ψ}{U_{0} L}, β^{*} = β L, \\ Q_{0}^{*} = \frac{Q_{0}}{U_{0} W L}, τ_{w}^{*} = \frac{τ_{w}}{µ (y) U_{0} / L}, u^{*} = \frac{u}{U_{0}}, v^{*} = \frac{v}{U_{0}}, H^{*} = \frac{H}{L}, \end{array}$ (16)

where U₀ is reference velocity, L is the length of slit, H is the height and W is the width of slit. Nondimensional form of Eqs (12-15) after releasing “*” take the following form

$\nabla^{4} ψ - β^{2} E^{2} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (17)

$\frac{\partial^{2} ψ}{\partial y^{2}} = 0, \frac{\partial ψ}{\partial x} = 0, a t y = 0,$ (18)

$\frac{\partial ψ}{\partial y} = 0, \frac{\partial ψ}{\partial x} = - V_{0}, a t y = H,$ (19)

$ψ (0, 0) = 0, a n d ψ (0, H) = \frac{Q_{0}}{2 W} .$ (20)

2.1.1. Solution of the problem

To get the solution of problem given in Eqs (17-20), Inverse method [12] is used which requires the following function.

$ψ (x, y) = x R (y) + T (y),$ (21)

where R(y) and T(y) represent unknown functions.

Invoking Eq (21) in Eqs (17-20), one can obtain the following set of equations

$\frac{d^{4} R}{d y^{4}} - 2 β \frac{d^{3} R}{d y^{3}} + β^{2} \frac{d^{2} R}{d y^{2}} = 0,$ (22)

$\frac{d^{4} T}{d y^{4}} - 2 β \frac{d^{3} T}{d y^{3}} + β^{2} \frac{d^{2} T}{d y^{2}} = 0,$ (23)

and boundary conditions are

$R (0) = 0, \frac{d^{2} R (0)}{d y^{2}} = 0,$ (24)

$R (H) = - V_{0}, \frac{d R (H)}{d y} = 0,$ (25)

$T (0) = 0, \frac{d^{2} T (0)}{d y^{2}} = 0,$ (26)

$T (H) = \frac{Q_{0}}{2}, \frac{d T (H)}{d y} = 0.$ (27)

Solving Eqs (22,23) under the boundary conditions (24-27), one can obtained the following expressions for R(y) and T(y)

$(y) = \frac{V_{0} (2 + e^{H β} y β (1 - H β) - e^{y β} (2 - y β))}{- 2 + e^{H β} (2 - H β (2 - H β))},$ (28)

and

$(y) = \frac{Q_{0} (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 4 + e^{H β} (4 - 2 H β (2 - H β))} .$ (29)

Following stream function can be found after using R(y) and T(y) in Eq. (21)

$ψ (x, y) = \frac{(Q_{0} - 2 V_{0} x) (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 4 + e^{H β} (4 - 2 H β (2 - H β))}$ (30)

2.1.2. Velocity distribution

With the aid of above stream function and Eq (9), one can get the next components of velocity

$u (x, y) = \frac{- (Q_{0} - 2 V_{0} x) β (1 - H β - e^{- (H - y) β} (1 - y β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (31)

and

$v (x, y) = \frac{V_{0} (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 2 + e^{H β} (2 - H β (2 - H β))} .$ (32)

Above equations show that horizontal velocity u(x, y) depends upon x and y, also horizontal velocity is maximum at y = 0 i.e.

$u_{m a x} = \frac{- (Q_{0} - 2 V_{0} x) β (1 - H β - e^{- H β})}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (33)

and v(x, y) is maximum at y = H.

$v_{m a x} = V_{0} .$ (34)

Flow rate across the slit can be calculated by the following formula

$Q (x) = 2 \int_{0}^{H} u (x, y) d y,$ (35)

$Q (x) = Q_{0} - 2 V_{0} x,$ (36)

which results from the inlet to the outlet of rectangular cross section.

2.1.3. Pressure

In biological movements pressure performs an important role therefore to find the pressure, Eqs. (9-10) with the help of Eq (31) and Eq (32) are reduced into the following form

$\frac{\partial p}{\partial x} = - β e^{- β y} (x \frac{d^{2} R}{d y^{2}} + \frac{d^{2} T}{d y^{2}}) + e^{- β y} (x \frac{d^{3} R}{d y^{3}} + \frac{d^{3} T}{d y^{3}}),$ (37)

$\frac{\partial p}{\partial y} = - e^{- β y} (\frac{d^{2} R}{d y^{2}} - 2 β \frac{d R}{d y}) .$ (38)

Using R(y) and T(y), Eqs (37,38) take the following form

$\frac{\partial p}{\partial x} = \frac{- (Q_{0} - 2 V_{0} x) β^{3}}{- 4 + e^{H β} (4 - 2 H β (2 - H β))},$ (39)

$\frac{\partial p}{\partial y} = \frac{V_{0} β^{2} (2 e^{- y β} (1 - H β) - e^{- H β} (2 - y β))}{2 - 2 e^{- H β} - H β (2 - H β)} .$ (40)

Integration of above expressions yield the following form

$p (x, y) = \frac{- V_{0} β (4 e^{- y β} (1 - H β) + e^{- H β} y β (4 - y β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)} + f (x),$ (41)

where f(x) i is given by as follows

$f (x) = \frac{- x (Q_{0} - V_{0} x) β^{3}}{- 4 + e^{H β} (4 - 2 H β (2 - H β))} + C,$ (42)

where C is an arbitrary factor. Inserting Eq (42) in Eq (41), we can compose the subsequent form of pressure

$p (x, y) - p (0, 0) = \frac{- β (4 (- 1 + e^{- y β}) V_{0} (1 - H β) + e^{- H β} β (4 V_{0} y + Q_{0} x β - V_{0} (x^{2} + y^{2}) β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (43)

where p(0, 0) shows pressure at creek of rectangular cross section.

The average pressure drop is defined as follows

$\overset{̅}{p} (x) = \frac{1}{H} \int_{0}^{H} (p (x, y) - p (0, 0)) d y .$ (44)

After inserting value of p(x, y) − p(0, 0) from Eq (43), the above equation takes the following form.

$\overset{̅}{p} (x) = \frac{12 V_{0} {(1 - H β)}^{2} + e^{- H β} (3 H Q_{0} x β^{3} + V_{0} (- 12 - H β (- 12 - 6 H β + (H^{2} + 3 x^{2}) β^{2})))}{12 e^{- H β} - 6 (2 - H β (2 - H β))} .$ (45)

Pressure drops over the length of rectangular cross section is given by the following formula

$Δ \bar{p} (L) = \bar{p} (0) - \bar{p} (L),$ (46)

which reduces to

$Δ \bar{p} (L) = \frac{- e^{- H β} L (- Q_{0} + L V_{0}) β^{3}}{4 - 4 e^{- H β} - 2 H β (2 - H β)} .$ (47)

2.1.4. Wall shear stress

Shear forces are very crucial in the viscous flow with variable viscosity and support to find the resistance on plane which are penned as follows

${τ_{w} |}_{y = H} = - µ (y) {(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})}_{y = H},$ (48)

Invoking Eqs (31,32) in above equation, we attained

${τ_{w} |}_{y = H} = \frac{H (Q_{0} - 2 V_{0} x) β^{3}}{4 - 4 e^{- H β} - 2 H β (2 - H β)} .$ (49)

2.1.5. Fractional reabsorption and leakage flux

The ratio of reabsorption to filtration rate can be measured by the following formula

$F A = \frac{Q (0) - Q (L)}{Q (0)} = \frac{2 V_{0} L}{Q_{0}}$ (50)

Leakage flux is given by the following formula

$q (x) = - \frac{d Q (x)}{d x} = 2 V_{0} .$ (51)

2.2. Linear viscosity

Case II: $µ (y) = 1 - β y$

In this case we will take the following viscosity

$µ (y) = 1 - β y, β > 0,$ (52)

After injecting Eq (52) and Eq (9) in Eqs (2,3), we get

$\frac{\partial p}{\partial x} = 2 (1 - β y) \frac{\partial^{3} ψ}{\partial x^{2} \partial y} + \frac{\partial}{\partial y} {(1 - β y) (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}})},$ (53)

$\frac{\partial p}{\partial y} = (1 - β y) \frac{\partial}{\partial x} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}}) + 2 \frac{\partial}{\partial y} {(1 - β y) (- \frac{\partial^{2} ψ}{\partial x \partial y})} .$ (54)

Reducing pressure gradient from Eqs (53,54), one can obtain the compatibility equation in terms of stream function

$(1 - β y) \nabla^{4} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (55)

where $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ .

There will be no change in Eq (55) after using non-dimensional quantities given in Eq (16)

2.2.1. Solution of the problem

Substituting Eq (21) into the Eq (55) to get

$(1 - β y) \frac{d^{4} R}{d y^{4}} - 2 β \frac{d^{3} R}{d y^{3}} = 0,$ (56)

$(1 - β y) \frac{d^{4} T}{d y^{4}} - 2 β \frac{d^{3} T}{d y^{3}} = 0.$ (57)

On solving Eqs (56,57) under the boundary conditions (24-27), we get the solution

$R (y) = \frac{V_{0} (- 2 y β \ln (- β + H β^{2}) - 2 \ln (1 - y β) - y β (2 + 2 H β - y β - 2 \ln (- β + y β^{2})))}{- H β (- 2 - H β) + 2 \ln (1 - H β)},$ (58)

and

$T (y) = \frac{Q_{0} (2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{- 2 H β (- 2 - H β) + 4 \ln (1 - H β)},$ (59)

Invoking R(y) and T(y) into Eq. (21) to get the following stream function

$ψ (x, y) = \frac{(2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{2 H β (2 + H β) + 4 \ln (1 - H β)} (Q_{0} - 2 V_{0} x) .$ (60)

2.2.2. Velocity field

With the aid of Eq (9) and Eq (60), flow field in axial and vertical direction are found as follows

$u (x, y) = \frac{- (Q_{0} - 2 V_{0} x) β (- (H - y) β - \ln (- β + H β^{2}) + \ln (- β + y β^{2}))}{H β (2 + H β) + 2 ln (1 - H β)},$ (61)

and

$v (x, y) = \frac{V_{0} (2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{H β (2 + H β) + 2 ln (1 - H β)} .$ (62)

Horizontal velocity is maximum at y = 0.

$u_{m a x} = \frac{- (Q_{0} - 2 V_{0} x) β (- H β + ln (- β) - \ln (- β + H β^{2}))}{H β (2 + H β) + 2 ln (1 - H β)},$ (63)

and v(x, y) is maximum at y = H.

$v_{m a x} = V_{0} .$ (64)

Flow rate within the four-sided slit is attained by using Eq (61) in Eq (35).

$Q (x) = Q_{0} - 2 V_{0} x .$ (65)

2.2.3. Pressure distribution

To find pressure, Eqs (53,54) can be composed in following manner

$\frac{\partial p}{\partial x} = - β (x \frac{d^{2} R}{d y^{2}} + \frac{d^{2} T}{d y^{2}}) + (1 - β y) (x \frac{d^{3} R}{d y^{3}} + \frac{d^{3} T}{d y^{3}}),$ (66)

$\frac{\partial p}{\partial y} = - (1 - β y) \frac{d^{2} R}{d y^{2}} + 2 β \frac{d R}{d y} .$ (67)

By using Eqs (58,59) in Eqs (66,67) to get

$\frac{\partial p}{\partial x} = \frac{(Q_{0} - 2 V_{0} x) β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)},$ (68)

$\frac{\partial p}{\partial y} = \frac{2 V_{0} β^{2} (- 2 H β + 3 y β - 2 \ln (- β + H β^{2}) + 2 \ln (- β + y β^{2}))}{H β (2 + H β) + 2 \ln (1 - H β)} .$ (69)

Integrating Eq (69) yields

$p (x, y) = \frac{(Q_{0} - V_{0} x) x β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)} + H (y),$ (70)

where H(y) is given by as follows

$H (y) = \frac{- V_{0} β (4 y β \ln (- β + H β^{2}) + 4 \ln (1 - y β) - y β (- 4 - 4 H β + 3 y β + 4 \ln (- β + y β^{2})))}{H β (2 + H β) + 2 \ln (1 - H β)} + C,$ (71)

where C is an arbitrary factor. Using Eq (71) in Eq (70), we can write the subsequent form of pressure

$p (x, y) - p (0, 0) = - β (- β (Q_{0} x β + V_{0} (- x^{2} β + y (- 4 - 4 H β + 3 y β))) + 4 V_{0} (\ln (1 - y β) - y β (- \ln (- β + H β^{2}) + \ln (- β + y β^{2})))) / H β (2 + H β) + 2 ln (1 - H β),$ (72)

where p(0, 0) shows the pressure at entrance of the slit.

Mean pressure drop can be find by substituting the value of p(x, y) − p(0, 0) in Eq (44) as follows

$\overset{̅}{p} (x) = V_{0} (\frac{1}{H} - 2 β) + \frac{(H^{2} V_{0} + x (Q_{0} - V_{0} x)) β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)} .$ (73)

To get the pressure drop over the length L, invoke the values of p(0) and p(L) in Eq (47).

$Δ \bar{p} (L) = - \frac{L (Q_{0} - L V_{0}) β^{3}}{H β (2 + H β) + 2 ln (1 - H β)} .$ (74)

2.2.4. Wall shear stress

Invoking Eqs (61,62) in Eq (48) to attain wall shear stress

${τ_{w} |}_{y = H} = - \frac{H (Q_{0} - 2 V_{0} x) β^{3}}{(1 - H β) (H β (2 + H β) + 2 \ln (1 - H β))} .$ (75)

3. Constant viscosity

Case III: If we apply the limit β → 0 on Eq 31, 32 and 43 and solving the resulting expressions by the L' Hospital rule, one can attain the results of constant viscosity µ which are same as mentioned in Ref. [12].

$ψ (x, y) = \frac{1}{2} (\frac{Q_{0}}{2} - V_{0} x) \frac{y}{H} (3 - {(\frac{y}{H})}^{2}),$ (76)

$u (x, y) = \frac{3 (Q_{0} - 2 V_{0} x)}{4 H} (1 - {(\frac{y}{H})}^{2}),$ (77)

$v (x, y) = \frac{V_{0}}{2} \frac{y}{H} (3 - {(\frac{y}{H})}^{2}),$ (78)

$p (x, y) - p (0, 0) = - \frac{3 µ (Q_{0} - V_{0} x) x}{2 H^{3}} - \frac{3 µ V_{0} y^{2}}{2 H^{3}} .$ (79)

4. Application to kidneys flow

We have utilized biological data of mammalian kidneys to find the values of V₀ and Δp in artificial renal tubule which is assumed to be the permeable channel. In this study length of artificial renal tubule (slit) is chosen to be L = 0.12cm and height of slit has considered as H = 0.0005cm, average viscosity µ = 6.9×10⁻³dynsec/cm², Q₀ = 1.25×10⁻⁴cm³/sec, β = 2.5×10⁻³/cm and W = 0.1 cm (as $H ≪ W$ ) from Zhang et al [24]. Table 1 shows the calculated values of mean pressure and reabsorption rate for the constant and variable viscosity. As cited in Refs. (Mehboob et. al. [21], Layton et al [22], Korkmaz et al [25], Herranz et al [26] and Tortora et al [27] in usual conditions glomerulus pressure is assumed to be about $45 m m H g (59995.07433675 d y n / c m^{2})$ which is above the average pressure than that observed in tubes elsewhere in the body. It has stated in Table 1 the reducing values of reabsorption rate causes to lower the fractional reabsorption, but the average pressure drop in the renal tubule enhances with the falling values of reabsorption rate, also the assumption of variable viscosity indicate that average pressure drop is elevated as matched with the case of constant viscosity.

The results of this study are beneficial to measure the change in pressure compared to the fractional reabsorption because the viscosity of the liquid decay near the boundary which is the consequence of lubricating nature of permeable wall. The numerical values given in table 1 indicate that the various rate of reabsorption requires the different mean pressure which may be supportive for the patients of nephrotic disorder. The present data from table 1 also support the person that how much intake of fluid is needed to generate the mean pressure next to the filtration rate which has examined by the blood CP and urine test.

Table 1. Fractional reabsorption, reabsorption rate of fluid and pressure variation for persistent (constant) viscosity and variable viscosity in a renal tubule.

Viscosity	FA	90%	80%	70%	60%
	V₀cm/sec	4.7×10⁻³	4.2×10⁻³	3.6×10⁻³	3.1×10⁻³
Constant	Δp(0.12)dyn/cm²	1212.19	2404.51	3835.3	5027.62
Variable	Δp(0.12)dyn/cm²	6460.62	7025.69	7703.78	8268.84

| Show Table

DownLoad: CSV

5. Results

Figure 2. Effect of V₀ at entrance region (x = 0.1).

DownLoad: Full-Size Img PowerPoint

Figure 3. Effect of β at entrance region (x = 0.1).

DownLoad: Full-Size Img PowerPoint

Figure 4. Effect of V₀ at middle region (x = 0.5).

DownLoad: Full-Size Img PowerPoint

Figure 5. Effect of β at middle region (x = 0.5).

DownLoad: Full-Size Img PowerPoint

Figure 6. Effect of V₀ at exit region (x = 0.9).

DownLoad: Full-Size Img PowerPoint

Figure 7. Effect of β at exit region (x = 0.9).

DownLoad: Full-Size Img PowerPoint

Figure 8. Effect of V₀ in the tranverse direction of flow.

DownLoad: Full-Size Img PowerPoint

Figure 9. Effect of β in the transverse direction of flow.

DownLoad: Full-Size Img PowerPoint

Figure 10. Effect of V₀ on pressure difference.

DownLoad: Full-Size Img PowerPoint

Figure 11. Effect of β on pressure difference.

DownLoad: Full-Size Img PowerPoint

Figure 12. Effect of V₀ on wall shear stress.

DownLoad: Full-Size Img PowerPoint

Figure 13. Effect of β on wall shear stress.

DownLoad: Full-Size Img PowerPoint

Figure 14. (a) V₀ = 0.4, (b) V₀ = 1.2, (c) V₀ = 1.5.

DownLoad: Full-Size Img PowerPoint

Figure 15. (a) β = 0.1, (b) β = 0.2, (c) β = 0.65.

DownLoad: Full-Size Img PowerPoint

6. Discussion

In this research the mathematical results of flow speed, flow pattern, pressure and shearing force are presented by the graphs which are plotted in software Mathematica under the effect of different values of reabsorption velocity V₀ and viscosity parameter β at x = 0.1, x = 0.5, x = 0.9, which are entry, central and leaving points of the rectangular cross section respectively.

Figures (2), (4) and (6) show the decaying effect of reabsorption rate V₀ on axial velocity at the points x = 0.1, 0.5, 0.9 on the slit and observe that decay is swifter near the center region (x = 0.5) but near the departure region (x = 0.9) back flow has been examined. It has been noted that near the centre of slit drift is highest due to change in pressure and near the borders of slit (y = h) the fluid movement become motionless due to deposition of the particles (reabsorption V₀).

Figures (3), (5) and (7) explain the decelerating impact of viscosity parameter β on axial rate near the center of the slit at the points x = 0.1, 0.5, 0.9 of the slit. The axial flow rises away from the centre (y = h) and decays near the centre line (y = 0) due to the growing values of viscosity parameter β, because resistance at the boundary wall (y = h) makes the motionless flow at the boundary but the resistive forces become stronger near the centre of slit (y = 0) due to pressure gradient $\frac{d p}{d x}$ which helps to make the decreasing effect on axial velocity.

Figure 8 illustrates the variation of reabsorption rate V₀ at the vertical direction of flow field, which displays the symmetry about the centre line (y = 0). The presence of reabsorption V₀ indicates the growing effect on the transverse flow near the permeable membrane of renal tube. Decaying effect of viscosity parameter β on the vertical flow is displayed through the Figure 9, it is observed that the resistive force retards the movement of the fluid molecules in vertical direction.

The pressure p plays a vital role for the flow of viscous fluid, therefore graphical results for pressure p are displayed in Figure 10 and Figure 11. The decreasing effect of reabsorption velocity V₀ on pressure difference is displayed in Figure 10 which shows that presence of reabsorption V₀ on the membrane require the low change in pressure for the viscous flow through permeable channel and the pressure at the entrance x = 0.1 is high as compared with the other region x = 0.9 of the slit. Figure 11 indicates that rising values of β help to reduce the pressure difference from entry point x = 0.1 to the centre line y = 0, which is evidenced that decreasing viscosity β help to reduce the pressure for the viscous fluid flow.

When the fluid is in contact with the surface then wall shear stress is an important phenomenon on the boundary surface. The variation in wall shear stress against V₀ is described in Figure 12 which shows the reduction in the shear stress on the wall with the mounting values of reabsorption V₀. The constant reabsorption V₀ on the wall helps to reduce the shearing forces on the permeable wall, which indicates that flow across the tube move easily without the large load. The rising values of parameter β results to improve the shearing forces on the wall because the growing values of β increases the resistance near the boundary wall which requires the high load for the fluid flow near the boundary.

The pattern of the flow field can be observed by the streamlines which are plotted in Figure 14 (a-c) and Figure 15 (a-c). It is examined that addition in reabsorption velocity V₀ triggers the number of lines which indicates that the fluid is becoming thin in the presence of reabsorption V₀, but the number of streamlines reduces with the mounting values of viscosity parameter β which indicates that fluid is going to be thin, and it contains more resistance for the fluid flow.

7. Conclusions

This research presents the mathematical modelling of plasma flow through the renal tube, the two-dimensional slit in rectangular coordinate is considered in the analogy of proximal renal tube and viscous fluid is assumed as a plasma with variable viscosity from centre line to the boundary surface due to reabsorption on the permeable wall. The complex mathematical model is solved by the inverse method and exact expression for flow speed, load, shear force and filtration are calculated. The application of the present study is presented with the numerical data present in the literature. The special cases for the linear and constant reabsorption are also obtained and observed that results of constant reabsorptions match with the existing results (Haroon et al. [12]) which validates the current research. The impact of reabsorption and viscosity parameter on the physical quantities of the flow field are observed through graphs. The numerical results of mean pressure and constant reabsorption against filtration rate show that constant viscosity of the plasma in slit requires less amount of average pressure but the biofluid having variable viscosity along the slit requires high average pressure for the fluid flow.

References

[1]	Robyt J (2001) Polysaccharides: energy storage. John Wiley & Sons.
[2]	Wang HM, Loganathan D, Linhardt RJ (1991) Determination of the pKa of glucuronic acid and the carboxy groups of heparin by 13C-nuclear-magnetic-resonance spectroscopy. Biochem J 278: 689-95. doi: 10.1042/bj2780689
[3]	Kyte J (1995) Structure in protein chemistry. New Yorg: Garland Publishing Inc.
[4]	Crouzier T, Boudou T, Picart C (2010) Polysaccharide-based polyelectrolyte multilayers. Curr Opin Colloid In 15: 417-426. doi: 10.1016/j.cocis.2010.05.007
[5]	Shukla RK (2011) Carbohydrate Molecules: An Expanding Horizon in Drug Delivery and Biomedicine. Crit Rev Ther Drug Carrier Syst 28: 255-292. doi: 10.1615/CritRevTherDrugCarrierSyst.v28.i3.20
[6]	Dumitriu S (2004) Polysaccharides: Structural Diversity and Functional Versatility. Second Edition Marcel Dekker: New York.
[7]	Andresen IL, Skipnes O, Smidsrod O, et al. (1977) Some biological functions of matrix components in benthic algae in relation to their chemistry and the composition of seawater. ACS SymSer 48: 361-381.
[8]	Sadoff HL (1975) Encystment and germination in Azotobactervinelandii. Bacteriol Rev 39: 516-539.
[9]	Haug A, Larsen B, Smidsrød O (1966) A Study of the Constitution of Alginic Acid by Partial Acid Hydrolysis. Acta Chem Scand 20: 183-190. doi: 10.3891/acta.chem.scand.20-0183
[10]	Simionescu CI, Popa VI, Rusan V, et al. (1976) The influence of structural units distribution on macromolecular conformation of alginic acid. Cell Chem Technol 10: 587-594.
[11]	Min KH, Sasaki SF, Kashiwabara Y, et al. (1977) Fine structure of SMG alginate fragment in the light of its degradation by alginate lyases of Pseudomonas sp. J Biochem 81: 555-562.
[12]	Harding SE, Varum KM, Stokke BT, et al. (1991) Molecular weight determination of polysaccharides. Adv Carbohydr Anal 1:63-144.
[13]	Martinsen A, Skjåk-Bræk G, Smidsrød O (1991) Comparison of different methods for determination of molecular weight and molecular weight distribution of alginates. Carbohydr Polym 15: 171-193. doi: 10.1016/0144-8617(91)90031-7
[14]	Smidsrød O, Haug A (1968) Dependence upon uronic acid composition of some ion-exchange properties of alginates. Acta Chem Scand 22:1989-1997. doi: 10.3891/acta.chem.scand.22-1989
[15]	Sikorski P, Mo F, Skjåk-Bræk G, et al. (2007) Evidence for Egg-Box-Compatible Interactions in Calcium−Alginate Gels from Fiber X-ray Diffraction. Biomacromolecules 8: 2098-2103. doi: 10.1021/bm0701503
[16]	Haug A, Smidsrod O (1970) Selectivity of some anionic polymers for divalent metal ions. Acta Chem Scand 24: 843-854. doi: 10.3891/acta.chem.scand.24-0843
[17]	Haug A, Myklestad S, Larsen B, et al. (1967) Correlation between chemical structure and physical properties of alginates. Acta Chem Scand 21: 768-778. doi: 10.3891/acta.chem.scand.21-0768
[18]	Grasselli M, Diaz LE, Cascone O (1993) Beaded matrices from cross-linked alginate for affinity and ion exchange chromatography of proteins. Biotechnol Tech 7: 707-712. doi: 10.1007/BF00152617
[19]	Pawar SN, Edgar KJ (2012) Alginate derivatization: a review of chemistry, properties and application. Biomaterials 33: 3279-3305. doi: 10.1016/j.biomaterials.2012.01.007
[20]	Hari PR, Chandy T, Sharma CP (1996) Chitosan/calcium alginate beads for oral delivery of insulin. J Appl Polym Sci 59: 1795-1801. doi: 10.1002/(SICI)1097-4628(19960314)59:11<1795::AID-APP16>3.0.CO;2-T
[21]	Huguet ML, Groboillot A, Neufeld RJ, et al. (1994) Hemoglobin encapsulation in chitosan/calcium alginate beads. J Appl Polym Sci 51: 1427-1432. doi: 10.1002/app.1994.070510810
[22]	MiFL, Sung HW, Shyu SS (2002) Drug release from chitosan-alginate complex beads reinforced by a naturally occurring crosslinking agent. Carbohydr Polym 48: 61-72. doi: 10.1016/S0144-8617(01)00212-0
[23]	Chan AW, Whitney RA, Neufeld R (2009) Semisynthesis of a controlled stimuli-responsive alginate hydrogel. Biomacromolecules 10: 609-616. doi: 10.1021/bm801316z
[24]	Bhattarai N, Zhang M (2007) Controlled synthesis and structural stability of alginate-based nanofibers. Nanotechnology 18: 455601-455611. doi: 10.1088/0957-4484/18/45/455601
[25]	Bouhadir KH, Hausman DS, Mooney DJ (1999) Synthesis of cross-linked poly(aldehyde guluronate) hydrogels. Polymer 40: 3575-3584. doi: 10.1016/S0032-3861(98)00550-3
[26]	Xu JB, Bartley JP, Johnson RA (2003) Preparation and characterization of alginate hydrogel membranes crosslinked using a water-soluble carbodiimide. J Appl Polym Sci 90: 747-753. doi: 10.1002/app.12713
[27]	Boontheekul T, Kong HJ, Mooney DJ (2005) Controlling alginate gel degradation utilizing partial oxidation and bimodal molecular weight distribution. Biomaterials 26: 2455-2465. doi: 10.1016/j.biomaterials.2004.06.044
[28]	Gomez CG, Rinaudo M, Villar MA (2007) Oxidation of sodium alginate and characterization of the oxidized derivatives. Carbohydr Polym 67: 296-304. doi: 10.1016/j.carbpol.2006.05.025
[29]	Kang HA, Jeon GJ, Lee MY, et al. (2002) Effectiveness test of alginate-derived polymeric surfactants. J Chem Technol Biot 77: 205-210. doi: 10.1002/jctb.550
[30]	Kang HA, Shin MS, Yang JW (2002) Preparation and characterization of hydrophobically modified alginate. Polym Bull 47: 429-435. doi: 10.1007/s002890200005
[31]	Laurienzo P, Malinconico M, Motta A, et al. (2005) Synthesis and characterization of a novel alginate-poly(ethylene glycol) graft copolymer. Carbohydr Polym 62: 274-282. doi: 10.1016/j.carbpol.2005.08.005
[32]	Skjåk-Bræk G, Zanetti F, Paoletti S (1989) Effect of acetylation on some solution and gelling properties of alginates. Carbohydr Res 185: 131-138. doi: 10.1016/0008-6215(89)84028-5
[33]	Coleman RJ, Lawrie G, Lambert LK, et al. (2011) Phosphorylation of alginate: synthesis, characterization, and evaluation of in vitro mineralization capacity. Biomacromolecules 12: 889-897. doi: 10.1021/bm1011773
[34]	Freeman I, Kedem A, Cohen S (2008) The effect of sulfation of alginate hydrogels on the specific binding and controlled release of heparin-binding proteins. Biomaterials 29: 3260-3268. doi: 10.1016/j.biomaterials.2008.04.025
[35]	Sand A, Yadav M, Mishra DK, et al. (2010) Modification of alginate by grafting of N-vinyl-2-pyrrolidone and studies of physicochemical properties in terms of swelling capacity, metal-ion uptake and flocculation. Carbohydr Polym 80: 1147-1154. doi: 10.1016/j.carbpol.2010.01.036
[36]	Sinquin A, Hubert P, Dellacherie E (1993) Amphiphilic derivatives of alginate: evidence for intra-and intermolecular hydrophobic associations in aqueous solution. Langmuir 9: 3334-3337. doi: 10.1021/la00036a002
[37]	Yang JS, Zhou QQ, He W (2013) Amphipathicity and self-assembly behavior of amphiphilic alginate esters. Carbohydr Polym 92: 223-227. doi: 10.1016/j.carbpol.2012.08.100
[38]	Mahou R, Meier RPH, Bühler LH, et al. (2014) Alginate-Poly(ethylene glycol) Hybrid Microspheres for Primary Cell Microencapsulation. Materials 7: 275-286. doi: 10.3390/ma7010275
[39]	Bernkop-Schnurch A, Kast CE, Richter MF (2001) Improvement in the mucoadhesive properties of alginate by the covalent attachment of cysteine. J Controlled Release 71: 277-285. doi: 10.1016/S0168-3659(01)00227-9
[40]	Jindal AB, Wasnik MN, Nair HA (2010) Synthesis of thiolated alginate and evaluation of mucoadhesiveness, cytotoxicity and release retardant properties. Indian J Pharm Sci 72: 766-774. doi: 10.4103/0250-474X.84590
[41]	Leone G, Torricelli P, Chiumiento A, et al. (2008) Amidic alginate hydrogel for nucleus pulposus replacement. J Biomed Mater Res A 84: 391-401.
[42]	Vallée F, Müller C, Durand A, et al. (2009) Synthesis and rheological properties of hydrogels based on amphiphilic alginate-amide derivatives. Carbohydr Res 344: 223-228. doi: 10.1016/j.carres.2008.10.029
[43]	Coates EE, Riggin CN, Fishe JP (2013) Photocrosslinked alginate with hyaluronic acid hydrogels as vehicles for mesenchymal stem cell encapsulation and chondrogenesis. J Biomed Mater Res A 101: 1962-1970.
[44]	Kumar MN, Muzzarelli RAA, Muzzarelli C, et al. (2004) Chitosan chemistry and pharmaceutical perspectives. Chem Rev 104: 6017-84. doi: 10.1021/cr030441b
[45]	Zhao Y, Park RD, Muzzarelli RAA (2010) Chitin Deacetylases: Properties and Applications. Mar Drugs 8: 24-46. doi: 10.3390/md8010024
[46]	Trzcinski S (2006) Combined degradation of chitosan. Polish Chitin Society Monograph XI, 103-11.
[47]	Varum KM, Ottoy MH, Smidsrød O (1994) Water solubility of partially N-acetylated chitosan as a function of pH: effect of chemical composition and depolymerization. Carbohydr Polyms 25: 65-70. doi: 10.1016/0144-8617(94)90140-6
[48]	Knaul JZ, Kasaai MR, Bui VT, et al. (1998) Characterization of deacetylated chitosan and chitosan molecular weight review. Can J Chem 76: 1699-1706.
[49]	Chattopadhyay DP, Inamdar MS (2010) Aqueous behavior of chitosan. Int Polym Science 2010 doi:10.1155/2010/939536.
[50]	Roberts GAF (1992) Chitin Chemistry. Houndmills UK: Macmillan Press.
[51]	An HK, Park BY, Kim DS (2001) Crab shell for the removal of heavy metals from aqueous solutions. Water Res 35: 3551-3556. doi: 10.1016/S0043-1354(01)00099-9
[52]	Wang QZ, Chen XG, Liu N, et al. (2006) Protonation constants of chitosan with different molecular weight and degree of deacetylation. Carbohydr Polym 65: 194-201. doi: 10.1016/j.carbpol.2006.01.001
[53]	Rinaudo M (2006) Chitin and chitosan: Properties and applications. Prog Polym Sci 31: 603-632. doi: 10.1016/j.progpolymsci.2006.06.001
[54]	Baxter A, Dillon M, Taylor KD, et al. (1992) Improved method for I.R. determination of the degree of N-acetylation of chitosan. Int J Biol Macromol 14: 166-169.
[55]	Muzzarelli RAA, Rochetti R (1985) Determination of the degree of acetylation of chitosans by first derivative ultraviolet spectrophotometry. Carbohydr Polym 5: 461-472. doi: 10.1016/0144-8617(85)90005-0
[56]	Vårum KM, Anthonsen MW, Grasdalen H, et al. (1991) Determination of the degree of N-acetylation and the distribution of N-acetyl groups in partially N-deacetylated chitins (chitosans) by high-field n.m.r. spectroscopy. Carbohydr Res 211:17-23. doi: 10.1016/0008-6215(91)84142-2
[57]	Värum KM, Anthonsen MW, Grasdalen H, et al. (1991) 13C-NMR studies of the acetylation sequences in partially N-deacetylated chitins (chitosans). Carbohydr Res 217: 19-27. doi: 10.1016/0008-6215(91)84113-S
[58]	Kasaai MR (2010) Determination of the degree of N-acetylation for chitin and chitosan by various NMR spectroscopy techniques: A review. Carbohydr Polym 79: 801-810. doi: 10.1016/j.carbpol.2009.10.051
[59]	Sreenivas SA, Pai KV (2008) ThiolatedChitosans: novel polymers for mucoadhesive drug delivery -A Review. Trop J Pharm Res 7: 1077-1088.
[60]	Jayakumar R, Nwe N, Tokura S, et al. (2007) Sulfated chitin and chitosan as novel biomaterials. Int J Biol Macromol 40: 175-181. doi: 10.1016/j.ijbiomac.2006.06.021
[61]	Sashiwa H, Kawasaki N, Nakayama A (2002) Synthesis of water-soluble chitosan derivatives by simple acetylation. Biomacromolecules 3: 1126-1128. doi: 10.1021/bm0200480
[62]	Jayakumar R, Reis RL, Mano JF (2006) Chemistry and Applications of Phosphorylated Chitin and Chitosan. e-Polymers 6: 447-462.
[63]	Jayakumar R, Nagahama H, Furuike T, et al. (2008) Synthesis of phosphorylated chitosan by novel method and its characterization. Int J Biol Macromol 42: 335-339. doi: 10.1016/j.ijbiomac.2007.12.011
[64]	Rúnarsson ÖV, Holappa J, Jónsdóttir S, et al. (2008) N-selective “one pot” synthesis of highly N-substituted trimethyl chitosan (TMC). Carbohydr Polym 74: 740-744. doi: 10.1016/j.carbpol.2008.03.008
[65]	Rosenthal R, Günzel D, Finger C, et al. (2012) The effect of chitosan on transcellular and paracellular mechanisms in the intestinal epithelial barrier. Biomaterials 33: 2791-2800. doi: 10.1016/j.biomaterials.2011.12.034
[66]	Muzzarelli RAA, Tanfani F, Emanuelli M, et al. (1982) N-(Carboxymethylidene) chitosans and N-(Carboxymethyl)-chitosans: Novel Chelating Polyampholytes obtained from Chitosan Glyoxylate. Carbohydr Res 107: 199-214. doi: 10.1016/S0008-6215(00)80539-X
[67]	Mourya VK, Inamdar NN, Tiwari A (2010) Carboxymethyl chitosan and its applications. Adv Mat Lett 1: 11-33. doi: 10.5185/amlett.2010.3108
[68]	George M, Abraham TE (2006) Polyionic Hydrocolloids forma the intestinal delivery of protein drugs: Alginate and chitosan-A review. J Controlled Release 114: 1-14. doi: 10.1016/j.jconrel.2006.04.017
[69]	Alves NM, Mano JF (2008) Chitosan derivatives obtained by chemical modification for biomedical and enviromental application. Int J Biol Macromol 43: 401-414. doi: 10.1016/j.ijbiomac.2008.09.007
[70]	Riva RL, Ragelle H, des Rieux A, et al. (2011) Chitosan and chitosan derivatives in drug delivery and tissue engineering. Adv Polym Sci 244: 19-44. doi: 10.1007/12_2011_137
[71]	Giri TK, Thakur A, Alexander A, et al. (2012) Modified chitosan hydrogels ad drug delivery and tissue engineering system: present status and application. Acta Pharm Sin B 2: 439-449. doi: 10.1016/j.apsb.2012.07.004
[72]	Berger J, Reist M, Mayer JM, et al. (2004) Structure and interactions in covalently and ionicallycrosslinked chitosan hydrogels for biomedical applications. Eur J Pharm Biopharm 57: 19-34. doi: 10.1016/S0939-6411(03)00161-9
[73]	Cai H, Zhang ZP, Sun PC, et al. (2005) Synthesis and characterization of thermo-and pH-sensitive hydrogels based on Chitosan-grafted N-isopropylacrylamide via γ-radiation. Rad Phys Chem 74: 26-30. doi: 10.1016/j.radphyschem.2004.10.007
[74]	El-Sherbiny IM, Smyth HDC (2010) Poly(ethylene glycol)-carboxymethyl chitosan-based pH-responsive hydrogels: photo-induced synthesis, characterization, swelling, and in vitro evaluation as potential drug carriers. Carbohydr Res 345: 2004-2012. doi: 10.1016/j.carres.2010.07.026
[75]	Yamada K, Chen T, Kumar G, et al. (2000) Chitosan Based Water-Resistant Adhesive. Analogy to Mussel Glue. Biomacromolecules 1: 252-8.
[76]	Il'ina AV, Varlamov VP (2005) Chitosan-based polyelectrolyte complexes: A review. Appl Biochem Micro 41: 5-11. doi: 10.1007/s10438-005-0002-z
[77]	Lu Y, Sun W, Gu Z (2014) Stimuli responsive nanomaterials for therapeutic protein delivery. J Controlled Release 194: 1-19. doi: 10.1016/j.jconrel.2014.08.015
[78]	Haug A, Larsen B, Smidsrød O (1963) The degradation of alginate at different pH values. Acta Chem Scand 17: 1466. doi: 10.3891/acta.chem.scand.17-1466
[79]	Timell TE (1964) The acid Hydrolysis of glycosides I. General conditions and the effect of the nature of the aglycone. Can J Chem 42: 1456-1472.
[80]	Haug A, Larsen B, Smidsrød O (1967) Alkaline degradation of alginate. Acta Chem Scand 21: 2859-2870. doi: 10.3891/acta.chem.scand.21-2859
[81]	Smidsrød O, Haug A, Larsen B (1967) Oxidative-reductive depolymerization: a note on the comparison of degradation rates of different polymers by viscosity measurements. Carbohydr Res 5: 482-485. doi: 10.1016/S0008-6215(00)81123-4
[82]	Holme HK, Foros H, Pettersen H, et al. (2001) Thermal depolymerization of chitosan chloride. Carbohydr Polym 46: 287-294. doi: 10.1016/S0144-8617(00)00332-5
[83]	Vårum KM, Ottøy MH, Smidsrød O (2001) Acid hydrolysis of chitosans. Carbohydr Polym 46: 89-98. doi: 10.1016/S0144-8617(00)00288-5
[84]	Menchicchi B, Fuenzalida JP, Bobbili KB, et al. (2014) Structure of chitosan determines its interactions with mucin. Biomacromolecules 15: 3550-3558. doi: 10.1021/bm5007954
[85]	Chen S, Cao Y, Ferguson LR, et al. (2013) Evaluation of mucoadhesive coatings of chitosan and thiolated chitosan for the colonic delivery of microencapsulated probiotic bacteria. J Microencapsu 30: 103-115. doi: 10.3109/02652048.2012.700959
[86]	Rubinstein A (2005) Colon drug delivery. Drug Discov Today: Technol 2: 33-37.
[87]	Vandamme TF, Lenourry A, Charrueau C, et al. (2002) The use of polysaccharides to target drugs to the colon. Carbohydr Polym 48: 219-231. doi: 10.1016/S0144-8617(01)00263-6
[88]	Wong TY, Preston LA, Schiller NL (2000) ALGINATE LYASE: Review of Major Sources and Enzyme Characteristics, Structure-Function Analysis, Biological Roles, and Applications. Ann Rev Microbiol 54: 289-340. doi: 10.1146/annurev.micro.54.1.289
[89]	Gacesa P (1988) Alginates. Carbohydr Polym 8: 161-182. doi: 10.1016/0144-8617(88)90001-X
[90]	Wargacki AJ, Leonard E, Win MN, et al. (2012) An Engineered Microbial Platform for Direct Biofuel Production from Brown Macroalgae. Science 335: 308-313. doi: 10.1126/science.1214547
[91]	Kean T, Thanou M (2010) Biodegradation, biodistribution and toxicity of chitosan. Adv Drug Deliv Rev 62: 3-11. doi: 10.1016/j.addr.2009.09.004
[92]	Funkhouser JD, Aronson NN (2007) Chitinase family GH18: evolutionary insights from the genomic history of a diverse protein family. BMC Evol Biol 7: 96. doi: 10.1186/1471-2148-7-96
[93]	Nordtveit RJ, Vårum KM, Smidsrød O (1994) Degradation of fully water-soluble, partially N-acetylated chitosans with lysozyme. Carbohydr Polym 23: 253-260. doi: 10.1016/0144-8617(94)90187-2
[94]	Brouwer J, van Leeuwen-Herberts T, Otting-van de Ruit M (1984) Determination of lysozyme in serum, urine, cerebrospinal fluid and feces by enzyme immunoassay. Clin Chim Acta 142: 21-30. doi: 10.1016/0009-8981(84)90097-4
[95]	Vårum KM, Myhr MM, Hjerde RJN, et al. (1997) In vitro degradation rates of partially N-acetylated chitosans in human serum. Carbohydr Res 299: 99-101. doi: 10.1016/S0008-6215(96)00332-1
[96]	Deckers D, Vanlint D, Callewaert L, et al. (2008) Role of the Lysozyme Inhibitor Ivy in Growth or Survival of Escherichia coli and Pseudomonas aeruginosa Bacteria in Hen Egg White and in Human Saliva and Breast Milk. Appl Environ Microbiol 74: 4434-4439. doi: 10.1128/AEM.00589-08
[97]	Callewaert L, Michiels CW (2010) Lysozyme in animal kingdom. J Biosci 35: 127-160. doi: 10.1007/s12038-010-0015-5
[98]	Cohen JS (1969) Proton Magnetic Resonance Studies of Human Lysozyme. Nature 223: 43-46. doi: 10.1038/223043a0
[99]	Kristiansen A, Vårum KM, Grasdalen H (1998) The interactions between highly de-N-acetylated chitosans and lysozyme from chicken egg white studied by 1H-NMR spectroscopy. Eur J Biochem 251: 335-342. doi: 10.1046/j.1432-1327.1998.2510335.x
[100]	Sasaki C, Kristiansen A, Fukamizo T, et al. (2003) Biospecific Fractionation of Chitosan. Biomacromolecules 4: 1686-1690. doi: 10.1021/bm034124q
[101]	Ren D, Yi H, Wang W, et al. (2005) The enzymatic degradation and swelling properties of chitosan matrices with different degrees of N-acetylaton. Carbohydr Res 340: 2403-2410. doi: 10.1016/j.carres.2005.07.022
[102]	Yang YM, Hu W, Wang XD, et al. (2007) The controlling biodegradation of chitosan fibers by n-acetylation in vitro and in vivo. J Mater Sci Mater Med 18: 2117-2121. doi: 10.1007/s10856-007-3013-x
[103]	Han T, Nwe N, Furuike T, et al. (2012) Methods of N-acetylated chitosan scaffolds and its in vitro biodegradation by lysozyme. J Biomed Sci Eng 5: 15-23. doi: 10.4236/jbise.2012.51003
[104]	Vårum KM, Holme HK, Izume M, et al. (1996) Determination of enzymatic hydrolysis specificity of partially N-acetylated chitosans. Biochem Biophy Acta (BBA) 1291: 5-15. doi: 10.1016/0304-4165(96)00038-4
[105]	Wen X, Kellum JA (2012) N-Acetyl-beta-D-Glucosaminidase (NAG). Encyclopedia Intensive Care Medicine 1509-1510.
[106]	Lim SM, Song DK, Oh SH, et al. (2008) In vitro and in vivo degradation behavior of acetylated chitosan porous beads. J Biomater Sci Polym Ed 19: 453-466.
[107]	Bajaj P, Schweller RM, Khademhosseini A, et al. (2014) 3D Biofabrication Strategies for Tissue Engineering and Regenerative Medicine. Annu Rev Biomed Eng 16: 247-276. doi: 10.1146/annurev-bioeng-071813-105155
[108]	Guarino V, Cirillo V, Altobelli R, et al. (2015) Polymer-based platforms by electric field-assisted techniques for tissue engineering and cancer therapy. Expert Rev Med Devices 12: 113-29. doi: 10.1586/17434440.2014.953058
[109]	Li Z, Zhang M (2005) Chitosan-alginate as scaffolding material for cartilage tissue engineering. J Biomed Mater Res A 75: 485-493.
[110]	Phan-Lai V, Florczyk SJ, Kievit FM (2013) Three-Dimensional Scaffolds to Evaluate Tumor Associated Fibroblast-Mediated Suppression of Breast Tumor Specific T Cells. Biomacromolecules 14: 1330-1337. doi: 10.1021/bm301928u
[111]	Florczyk SJ, Liu G, Kievit FM, et al. (2012) 3D porous chitosan-alginate scaffolds: a new matrix for studying prostate cancer cell-lymphocyte interactions in vitro. Adv Healthc Mater 1: 590-599. doi: 10.1002/adhm.201100054
[112]	Kievit FM Florczyk SJ, Leung MC, et al. (2010) Chitosan-alginate 3D scaffolds as a mimic of the glioma tumor microenvironment. Biomaterials 31: 5903-5910. doi: 10.1016/j.biomaterials.2010.03.062
[113]	Mura S, Nicolas J, Couvreur P (2013) Stimuli-Responsive Nanocarriers for Drug Delivery. Nat Mater 12: 991-1003. doi: 10.1038/nmat3776
[114]	Gonçalves M, Figueira P, Maciel D (2014) pH-sensitive Laponite®/doxorubicin/alginate nanohybrids with improved anticancer efficacy. Acta Biomater 10: 300-307 doi: 10.1016/j.actbio.2013.09.013
[115]	Kim CK, Lee EJ (1992) The controlled release of blue dextran from alginate beads. Int J Pharm 79: 11-19. doi: 10.1016/0378-5173(92)90088-J
[116]	Gulbake A, Jain SK (2012) Chitosan: a potential polymer for colon-specific drug delivery system. Exp Opin Drug Deliv 9: 713-729. doi: 10.1517/17425247.2012.682148
[117]	Ganguly K, Aminabhavi TM, Kulkarni AR (2011) Colon Targeting of 5-Fluorouracil Using Polyethylene Glycol Cross-linked Chitosan Microspheres Enteric Coated with Cellulose Acetate Phthalate. Ind Eng Chem Res 50: 11797-11807. doi: 10.1021/ie201623d
[118]	Lai CK, Lu YL, Hsieh JT, et al. (2014), Development of chitosan/heparin nanoparticle-encapsulated cytolethal distending toxin for gastric cancer therapy. Nanomedicine 9: 803-817.
[119]	Arora S, Budhiraja RD (2012) Chitosan-alginate microcapsules of amoxicillin for gastric stability and mucoadhesion. J Adv Pharm Techno Res 3: 68-74.
[120]	Du J, Dai J, Liu JL, et al. (2006) Novel pH-sensitive polyelectrolyte carboxymethyl Konjac glucomannan-chitosan beads as drug carriers. React Funct Polym 66: 1055-1061. doi: 10.1016/j.reactfunctpolym.2006.01.014
[121]	Liu Z, Jiao Y, Liu F, et al. (2007) Heparin/chitosan nanoparticle carriers prepared by polyelectrolyte complexation. J Biomed Mater Res Part A 83: 806-812.
[122]	Zhou T, Xiao C, Fan J, et al. (2012) A nanogel of on-site tunable pH-response for efficient anticancer drug delivery. Acta Biomater 9: 4546-4557.
[123]	Kutlu C, Çakmak AS, Gümüşderelioğlu M (2014) Double-effective chitosan scaffold-PLGA nanoparticle system for brain tumour therapy: in vitro study. J Microencapsul 31: 700-707. doi: 10.3109/02652048.2014.913727
[124]	Feng C, Song R, Sun G, et al. (2014) Immobilization of Coacervate Microcapsules in Multilayer Sodium Alginate Beads for Efficient Oral Anticancer Drug Delivery. Biomacromolecules15: 985-996.
[125]	Anchisi C, Meloni MC, Maccioni AM (2006) Chitosan beads loaded with essential oils in cosmetic formulations. J Cosmet Sci 57: 205-214.
[126]	Ariza-Avidad M, Nieto A, Salinas-Castillo A, et al. (2014) Monitoring of degradation of porous silicon photonic crystals using digital photograph. Nanoscale Res Lett 9: 410. doi: 10.1186/1556-276X-9-410
[127]	Jung SM, Yoon GH, Lee HC, et al. (2015) Chitosan nanoparticle/PCL nanofiber composite for wound dressing and drug delivery. J Biomater Sci Polym 26: 252-263. doi: 10.1080/09205063.2014.996699
[128]	Badawy ME, El-Aswad AF (2014) Bioactive paper sensor based on the acetylcholinesterase for the rapid detection of organophosphate and carbamate pesticides. Int J Anal Chem 2014 Article ID 536823.
[129]	Spreen KA, Zikakis JP, Austin PR (1984) Chitin, Chitosan and Related Enzymes. In J P Zikakis (Ed.) Academic Press, Orlando, 57.
[130]	Vázquez N, Chacón M, Meana Á, et al. (2015) Keratin-chitosan membranes as scaffold for tissue engineering of human cornea. Histol Histopathol 30: 813-821.
[131]	Cheng YH, Hung KH, Tsai TH, et al. (2014) Sustained delivery of latanoprost by thermosensitive chitosan-gelatin-based hydrogel for controlling ocular hypertension. Acta Biomater 10: 4360-4366. doi: 10.1016/j.actbio.2014.05.031
[132]	Wu FC, Tseng RL, Juang RS (2010) A review and experimental verification of using chitosan and its derivatives as adsorbents for selected heavy metals. J Environ Manage 91: 798-806. doi: 10.1016/j.jenvman.2009.10.018
[133]	Szymańska E, Winnicka K, AmelianA, et al. (2014) Vaginal chitosan tablets with clotrimazole-design and evaluation of mucoadhesive properties using porcine vaginal mucosa, mucin and gelatin. Chem Pharm Bull (Tokyo) 62:160-167. doi: 10.1248/cpb.c13-00689
[134]	Rao TV, Kumar GK, Ahmed MG, et al. (2012) Development and evaluation of chitosan based oral controlled matrix tablets of losartan potassium. Int J Pharm Investig 2: 157-161. doi: 10.4103/2230-973X.104399
[135]	Wassmer S, Rafat M, Fong WG, et al. (2013) Chitosan microparticles for delivery of proteins to the retina. Acta Biomater 9: 7855-7864. doi: 10.1016/j.actbio.2013.04.025
[136]	Chen CK, Wang Q, Jones CH, et al. (2014) Synthesis of pH-responsive chitosan nanocapsules for the controlled delivery of doxorubicin. Langmuir 30: 4111-4119. doi: 10.1021/la4040485
[137]	Prasad M, Palanivelu P (2014) Immobilization of a thermostable, fungal recombinant chitinase on biocompatible chitosan beads and the properties of the immobilized enzyme. Biotechnol Appl Biochem 62: 523-529.
[138]	Jain SK, Chourasia MK, Sabitha M, et al. (2003) Development and characterization of transdermal drug delivery systems for diltiazem hydrochloride. Drug Deliv 10: 169-177. doi: 10.1080/713840400
[139]	Kong M, Chen XG, Xing K, et al. (2010) Antimicrobial properties of chitosan and mode of action: a state of the art review. Int J Food Microbiol 144: 51-63. doi: 10.1016/j.ijfoodmicro.2010.09.012
[140]	Yang J, Luo K, Li D, et al. (2013) Preparation, characterization and in vitro anticoagulant activity of highly sulfated chitosan. Int J Biol Macromol 52: 25-31. doi: 10.1016/j.ijbiomac.2012.09.027
[141]	Whang HS, Kirsch W, Zhu YH, et al. (2005) Hemostatic Agents Derived from Chitin and Chitosan. J Macromol Sci Part C: Polym Rev 45: 309-323. doi: 10.1080/15321790500304122
[142]	Čopíková J, Taubner T, Tůma J, et al. (2015) Cholesterol and fat lowering with hydrophobic polysaccharide derivatives. Carbohydr Polym 116: 207-214. doi: 10.1016/j.carbpol.2014.05.009
[143]	Pereira GG, Guterres SS, Balducci AG, et al. (2014) Polymeric films loaded with vitamin E and aloe vera for topical application in the treatment of burn wounds. Biomed Res Int 2014 ID 641590.
[144]	Rojas-Graü MA, Tapia MS, Rodríguez FJ, et al. (2007) Alginate and gellan based edible coatings as support of antibrowning agents applied on fresh-cut Fuji apple. Food Hydrocolloids 21: 118-127. doi: 10.1016/j.foodhyd.2006.03.001
[145]	Wargacki AJ, Leonard E, Win MN, et al. (2012) An engineered microbial platform for direct biofuel production from brown macroalgae. Science 335: 308-313. doi: 10.1126/science.1214547
[146]	Sarei F, Dounighi NM, Zolfagharian H, et al. (2013) Alginate Nanoparticles as a Promising Adjuvant and Vaccine Delivery System. Indian J Pharm Sci 75: 442-449. doi: 10.4103/0250-474X.119829
[147]	Tønnesen HH, Karlsen J (2002) Alginate in Drug Delivery Systems. Drug Dev Ind Pharm 28: 621-630. doi: 10.1081/DDC-120003853
[148]	Giri TK, Thakur D, Alexander A, et al. (2012) Alginate based Hydrogel as a Potential Biopolymeric Carrier for Drug Delivery and Cell Delivery Systems: Present Status and Applications. Curr Drug Deliv 9: 539-555. doi: 10.2174/156720112803529800
[149]	Murata Y, Sasaki N, Miyamoto E, et al. (2000) Use of floating alginate gel beads for stomach-specific drug deliver. Eur J Pharm Biopharm 50: 221-226. doi: 10.1016/S0939-6411(00)00110-7
[150]	Paul W, Sharma CP (2004) Chitosan and Alginate Wound Dressings: A Short Review. Trends Biomater Artif Organs 18: 18-23.
[151]	Tanga M, Chena W, Weira MD, et al. (2012) Human embryonic stem cell encapsulation in alginate microbeads in macroporous calcium phosphate cement for bone tissue engineering. Acta Biomater 8: 3436-3445. doi: 10.1016/j.actbio.2012.05.016
[152]	Martinez JC, Kim JW, Ye C, et al. (2012) A Microfluidic Approach to Encapsulate Living Cells in Uniform Alginate Hydrogel Microparticles. Macromol Biosci 12:946-951. doi: 10.1002/mabi.201100351
[153]	Yang Q, Lei S (2014) Alginate Dressing Application in Hemostasis After Using Seldinger Peripherally Inserted Central Venous Catheter in Tumor Patients. Indian J Hematol Blood Transfus DOI 10.1007/s12288-014-0490-1.
[154]	Kimura Y, Watanabe K, Okuda H (1996) Effects of soluble sodium alginate on cholesterol excretion and glucose tolerance in rats. J Ethnopharmacol 54: 47-54. doi: 10.1016/0378-8741(96)01449-3
[155]	Ueno M, Tamura Y, Toda N, et al. (2012) Sodium Alginate Oligosaccharides Attenuate Hypertension in Spontaneously Hypertensive Rats Fed a Low-Salt Diet. Clin Exp Hypertens 34: 305-310. doi: 10.3109/10641963.2011.577484
[156]	Odunsi ST, Vázquez-Roque MI, Camilleri M (2009) Effect of alginate on satiation, appetite, gastric function, and selected gut satiety hormones in overweight and obesity. Obesity 18: 1579-1584.
[157]	Bhunchu S, Rojsitthisak P (2014) Biopolymeric alginate-chitosan nanoparticles as drug delivery carriers for cancer therapy. Pharmazie 69: 563-570.
[158]	Idota Y, Harada H, Tomono T, et al. (2013) Alginate Enhances Excretion and Reduces Absorption of Strontium and Cesium in Rats. Biol Pharm Bull 36: 485-491. doi: 10.1248/bpb.b12-00899

This article has been cited by:

José H Valladares Pérez, Roberto Márquez Islas, Celia Sánchez Pérez, Augusto García Valenzuela, Measurement of resistivity of biofluids in the presence of an electrical double layer, 2025, 36, 0957-0233, 025113, 10.1088/1361-6501/adaa94

Reader Comments

Your name:*

Email:*
© 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Materials Science

1.8 3.7

Metrics

Article views(20757) PDF downloads(2665) Cited by(97)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Materials Science

Degradation properties and metabolic activity of alginate and chitosan polyelectrolytes for drug delivery and tissue engineering applications

Related Papers:

Abstract

1. Introduction

2. Materials and methods

2.1. Exponential viscosity

2.1.1. Solution of the problem

2.1.2. Velocity distribution

2.1.3. Pressure

2.1.4. Wall shear stress

2.1.5. Fractional reabsorption and leakage flux

2.2. Linear viscosity

2.2.1. Solution of the problem

2.2.2. Velocity field

2.2.3. Pressure distribution

2.2.4. Wall shear stress

3. Constant viscosity

4. Application to kidneys flow

5. Results

6. Discussion

7. Conclusions

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Materials and methods

2.1. Exponential viscosity

2.1.1. Solution of the problem

2.1.2. Velocity distribution

2.1.3. Pressure

2.1.4. Wall shear stress

2.1.5. Fractional reabsorption and leakage flux

2.2. Linear viscosity

2.2.1. Solution of the problem

2.2.2. Velocity field

2.2.3. Pressure distribution

2.2.4. Wall shear stress

3. Constant viscosity

4. Application to kidneys flow

5. Results

6. Discussion

7. Conclusions

References